To find a point on the line joining the points [tex]\((4, 12)\)[/tex] and [tex]\((-3, 5)\)[/tex] whose [tex]\(x\)[/tex]-coordinate is [tex]\(2\)[/tex], follow these steps:
1. Determine the slope [tex]\((m)\)[/tex] of the line passing through the points [tex]\((4, 12)\)[/tex] and [tex]\((-3, 5)\)[/tex]:
The formula for the slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in our points:
[tex]\[
m = \frac{5 - 12}{-3 - 4} = \frac{-7}{-7} = 1
\][/tex]
2. Calculate the y-intercept [tex]\((b)\)[/tex] of the line using the point-slope form of the equation of a line:
The point-slope form of the line's equation is:
[tex]\[
y = mx + b
\][/tex]
Substitute one of the points, for example, [tex]\((4, 12)\)[/tex], into the equation to solve for [tex]\(b\)[/tex]:
[tex]\[
12 = 1 \cdot 4 + b
\][/tex]
Simplify to solve for [tex]\(b\)[/tex]:
[tex]\[
12 = 4 + b \implies b = 12 - 4 = 8
\][/tex]
3. Now, use the slope and y-intercept to form the equation of the line:
[tex]\[
y = 1 \cdot x + 8 \implies y = x + 8
\][/tex]
4. Find the y-coordinate when the x-coordinate is 2:
Substitute [tex]\(x = 2\)[/tex] into the equation of the line:
[tex]\[
y = 2 + 8 = 10
\][/tex]
So, the coordinates of the point on the line with an [tex]\(x\)[/tex]-coordinate of [tex]\(2\)[/tex] are [tex]\((2, 10)\)[/tex].
The slope [tex]\(m\)[/tex] is [tex]\(1\)[/tex], the y-intercept [tex]\(b\)[/tex] is [tex]\(8\)[/tex], and the [tex]\(y\)[/tex]-coordinate at [tex]\(x = 2\)[/tex] is [tex]\(10\)[/tex]. Thus, the point on the line is [tex]\((2, 10)\)[/tex].
